Applied Mechanics - Newton's Law
Newton's Laws of Motion
First Law: A body at rest or in uniform motion will remain at rest or in uniform motion unless
some external force is applied to it
Second Law of Motion: When a body is acted upon by a constant force, its resulting
acceleration is proportional to the force and inversely proportional to the mass,
Where,
a=acceleration, m/s2
F=force, N
M=mass of a body, kg
Third Law of Motion: It states that to every action force there is an equal and opposite
reaction force.
Motion:
Displacement
Velocity
Acceleration
Where, S=distance covered by a moving body in time t, m
V=Velocity of a moving body, m/s
A =acceleration of a moving body, m/s2
V0= initial velocity of a moving body, m/s
T=time of movement, s
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Newton’s Law of Gravitation
Any two bodies attract each other with a force that is proportional to the product of their
masses and inversely proportional to the square of the distance between them
where,
F=force of attraction, N
m1=mass of body one, kg
m2=mass of body second, kg
d=distance between two bodies, m
G=Newtonian constant of gravitation
Inertia
The inertia of a body may be defined as that property of a body which tends to resist a change in its state of rest or
motion.
Mass is defined as a quantitative measure of inertia.
Moment of Inertia of Areas
where,
Ix=moment of inertia of cross-sectional area about X-axis
Iy=moment of inertia of cross-sectional area about Y-axis
Iz=moment of inertia of cross-sectional area about Z-axis.
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Moment of inertia of areas
Mass moment of inertia (Im) of a body is given by:
Mass moment of inertia for different shapes of body
Rectangle
Triangle
Circle
Ellipse
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Circular Cylindrical Shell
where m= mass, r = radius
Right Circular Cylinder
where m= mass, r = radius
Semi cylinder
where m= mass, r = radius
Hemisphere
where m= mass, r = radius
Rectangular Parallelopiped
Uniform Slender Rod
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Right Circular Cone
Elliptical Cylinder
Hemispherical Shell
Torus (complete)
Applied Mechanics – Density
Density
Where,
ρ=density, g/cm3
Ρw=weight density, N/cm3
M=mass, g
V=volume, cm3
W=weight, N
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Applied Mechanics – Vibrations
Vibrations
1. Simple Harmonic Motion
where,
T=period of a vibration, s
n=frequency or vibration per unit time, 1/s
2. Spring Pendulum
where,
T=period, s
M=mass of pendulum
K=spring
3. Simple Pendulum
where,
l=length of the pendulum
g=acceleration due to gravity
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4. Wavelength
where,
V=total distance traveled in one second
λ=length of one wave
η=number of waves per second
5. Speed of sound
where,
V=speed of sound at temperature tc˚C, m/s
Vo=speed at 0˚C, m/s
tc=temperature, ˚C.
6. Beat Notes
N=n2-n1
where,
N=beat frequency, i.e., number of beats per second
N1, n2=frequencies of two sources producing the sound, vibrations/s
7. Doppler Effect
where,
No=frequency heard by the observer
ns=frequency of the source
V=velocity of sound
Vs=velocity of source
Vo=velocity of the observer
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8. Intensity of sound
where,
E=intensity of sound at any distance d, microwatts/cm2 or decibels
Eo=intensity of sound at unit distance, decibels
9. Vibrating Strings
where,
V=velocity of sound, m/s
N=frequency or number of waves passing by per second
λ=length of one wave or wavelength
F=tension in a rope or string, N
M=mass of string per unit length, kg/m
L=distance between two consecutive nodes, m
10. Sound Wave Through Gas
where,
V=wave velocity, cm/s
P=gas pressure, dynes/cm2
ρ=gas density, g/cm3
K=proportionality constant
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